23-Chem-A2 Unit Operations and Separation Processes · December 2015
Question 4 of 6: Natural-Convection Heat Transfer to a Long Square Duct
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
Paper format. National Exams — 04-CHEM-A2 Mechanical and Thermal Operations, December 2015. 3 hours, open book (one text). Six problems (Section A Mechanical Operations: A1–A3; Section B Thermal Operations: B1–B3), each 25 marks; candidates attempt at least two from each section. All six are worked below for completeness.
Reference texts. Coulson & Richardson, Chemical Engineering Vol. 1 (fluid flow, heat transfer) and Vol. 2 (particle technology, filtration, evaporation); McCabe, Smith & Harriott, Unit Operations of Chemical Engineering (7th ed.); Incropera & DeWitt, Fundamentals of Heat and Mass Transfer (8th ed.); Perry's Chemical Engineers' Handbook (9th ed.).
Note on the figureThe only figure in the paper is the LMTD correction-factor chart on page 5 (used in B3). It is read at the plotted parameters $P=0.25$, $R=2.0$; the value obtained ($F\approx0.94$) is confirmed analytically from the closed-form 1-outer-pass / 2-tube-pass expression.
Question B1: Natural-Convection Heat Transfer to a Long Square Duct (25 marks)
Given. A cold square duct in warmer still air; the tabulated air properties are evaluated at the film temperature and the compound buoyancy group is supplied directly.
Quantity
Value
Duct surface temperature
15 °C
Air temperature
39 °C
Side length $a$
32 cm = 0.32 m
Film temperature $T_f$
300 K
Air conductivity $\kappa$ @300 K
$26.14\times10^{-3}$ W/m·K
Buoyancy group $g\beta/(\nu\alpha)$
$94.1\times10^{6}$ m⁻³K⁻¹
Find. The natural-convection heat transfer rate per unit length of duct, $q'$ (W/m).
Figure B1 — Four-plate model of the duct: two vertical sides, a stably-stratified top and an unstable bottom (orientations inverted because the duct is cold). Heat flows from the warm air into the cold surfaces.
Approach. Decompose the square prism into four plates (two vertical sides, top, bottom), each with its own natural-convection correlation; evaluate $Ra$ from the supplied buoyancy group and sum the per-metre duties.
Film state and Rayleigh number. $T_f=\tfrac12(15+39)=27\,{}^\circ$C$=300$ K and $\Delta T=24$ K. The supplied group folds four properties into one, so $$Ra_L=\Big[\tfrac{g\beta}{\nu\alpha}\Big]\Delta T\,L^3=(94.1\times10^{6})(24)\,L^3.$$
Vertical sides ($L=a=0.32$ m). $$Ra_{\text{side}}=(94.1\times10^{6})(24)(0.32)^3=7.40\times10^{7},\quad Nu=0.59\,Ra^{1/4}=54.7,$$ $$h_{\text{side}}=\frac{Nu\,\kappa}{a}=\frac{54.7(0.02614)}{0.32}=4.47\ \text{W/m}^2\text{K}.$$
Horizontal faces ($L_c=a/2=0.16$ m). Using the plate length $A/P\approx a/2$ for a long strip, $Ra_{L_c}=(94.1\times10^{6})(24)(0.16)^3=9.25\times10^{6}$. Because the duct is cold, the top is stably stratified (weak) and the bottom unstable (strong): $$h_{\text{top}}=\frac{0.27\,Ra_{L_c}^{1/4}\kappa}{L_c}=2.43,\qquad h_{\text{bot}}=\frac{0.54\,Ra_{L_c}^{1/4}\kappa}{L_c}=4.87\ \text{W/m}^2\text{K}.$$
Heat transfer per unit length. Each face presents $a\times1=0.32$ m² per metre; summing the four: $$q'=\Delta T\,a\big(2h_{\text{side}}+h_{\text{top}}+h_{\text{bot}}\big)=24(0.32)\big[2(4.47)+2.43+4.87\big].$$ $q' \approx 125\ \text{W/m}$
Check (assumptions)Constant properties at $T_f$; laminar boundary layers (confirmed by $Ra<10^9$); radiation neglected; end effects ignored on the long duct. Radiation between a 15 °C surface and 39 °C surroundings would add a comparable term if surface emissivity were specified — here only convection is requested.
